The IRS, and most states, consider unemployment payments as taxable income, which means that you have to pay tax on these payments and report them on your return. If your sole source of income was from unemployment, and that income was less than the taxable threshold after exemptions and deductions, you might not owe any taxes. See this chart to determine your situation (note the exceptions as well).
Remember, if you don't owe any taxes and you had no withholding (that you would need to request a refund), then you may not be required to file a federal tax return, based on the thresholds in the chart linked to above. This may also mean you aren't required to file a state return, but requirements vary from state to state.
Unemployment income is reported on Form 1099-G. While entering this income in TurboTax, you'll be asked if you had any income withheld for tax purposes.
After you enter your unemployment income (along with any other sources of income, in the appropriate sections of TurboTax), we'll figure out how much (if any) is taxable by the IRS and your state.